Some properties about the eigenvalues of the distance matrix are already known. For example, in [1]}, the author characterizes the graphs with minimal spectral radius of the distance matrix in three classes of simple connected graphs with \(n\)  vertices: with fixed vertex connectivity, matching number and chromatic number, respectively. In [2]}, the authors characterize all connected graphs with  \(\partial _{n}(G)=-2\) . Furthermore, they characterize all connected graphs of diameter 2 with exactly three distance eigenvalues when  \(\partial _{1}(G)\)  is not an integer. They also conjecture that the complete \(k\) -partite graph is determined by its distance spectrum. Later, in [3]}, the author determines all graphs which satisfy  \(\partial _{n}(G)\in [-2.383 \ , \ 0]\) . A very complete survey of the state of the art on distance matrices up to 2014 appears in [4]} (see also
[5]}).
